## Banach and Hilbert spaces of vector-valued functions: their general theory and applications to holomorphy |

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### Contents

Hilbert spaces of functions wth values in a Hilbert space | 12 |

The multiplication operator ovir function Hilbert spaces | 22 |

The dual theory for conjugations | 30 |

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### Common terms and phrases

A X A to CL(N1,N1 A,N function Hilbert adjoint applications Bergman kernel Bergman space Cauchy sequence Chapter closed linear operator compact conditions are equivalent conjugations continuous linear operators Cq be open defined Definition disk dual EF(A F1 to F2 F2 to F1 following conditions function Banach space function Hilbert space functions on Q H_,K Hardy spaces Hartog Theorem Hence Hol(Q Hol(Q,N holomorphic functions inner product involution isometry kernel on A X A Kernel Theorem Lebesgue measure Lemma let f multiplication operator MY(MY Nonlinear partial differential norm obviously open set Q operator-valued kernel ordinary reproducing kernel partial differential equations PD kernel PD on A X A polydisk positive-definiteness Proposition Q E Cq Q X Q Question 1.4 reproducing variety single-valued space F space over F subset sup Y(z Theorem 3.4 theory Triviality VA,A vf e F Vz E Q